Anon-dualsignaturemethodfortheassignment ... · [6, 15, 17, 21], primai simplex [5, 13, 18] and...

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KONSTANTINOS PAPARRIZOSA non-dual signature method for the assignmentproblem and a generalization of the dual simplexmethod for the transportation problemRevue française d’automatique, d’informatique et de rechercheopérationnelle. Recherche opérationnelle, tome 22, no 3 (1988),p. 269-289.<http://www.numdam.org/item?id=RO_1988__22_3_269_0>

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Recherche opérationnelle/Opérations Research(vol. 22, n° 3, 1988, p. 269 à 289)

A NON-DUAL SIGNATURE METHOD FOR THE ASSIGNMENTPROBLEM AIMD A GENERALIZATION OF THE DUAL SIMPLEX

METHOD FOR THE TRANSPORTATION PROBLEM (*)

by Konstantinos PAPARRIZOS (*)

Abstract. — A combination of a new kind of dual relaxation with the signature idea provides anon-dual signature method for the assignment problem. The dual relaxation is also extended totransportation problems.

The algorithms, which consist of consequitive stages each one being a set of simplex type pivots,can be considered as a generalization of the dual simplex method. From the computational pointofview, the algorithms possess a new monotonie property, namely, during a stage ofthe algorithmsthe per unit improvement of the objective function is non-decreasing.

Keywords : Assignment problem, Transportation problem, Dual simplex method, Signatures,Rvoting.

Résumé. — Une combinaison d'un nouveau type de relaxation duel avec Vidée de la signaturefournit une non-duel méthode de signature pour le problème d'assignation. La relaxation duel seprolonge aussi au problème de transportation.

Les algorithmes qui sont formés par des stages consécutives dont chacun d'eux étant un ensemblesimplex de type de pivot, peuvent être considérer comme une generalization de la méthode desimplex duel Du point de vue computationy les algorithmes possèdent une nouvelle propriétémonotone, autrement dit, durant un stage des algorithmes l'amélioration par unité de la fonctionobjective est non décroissante.

Mots clés : Problème d'assignation, problème de transportation, méthode de simplex duel,signatures, pivotation.

1. INTRODUCTION

The assignment problem is perhaps the most studied and well solvedproblem in mathematical programming and numerous algorithms have beendiscovered to solve it in polynomial time. It has been generalized tobottleneck, quadratic and algebraic cases and has many applications. Somealgorithms, initially constructed for the assignment problem, have been

(*) Receivedl1) Democritus University of Thrace, Department of Mathematics, 67100 Xanthi, Greece.

Recherche opérationnelle/Opérations Research, 0399-0559/88/03 269 21/$ 4.10© AFCET-Gauthier-Villars

270 K. PAPARRIZOS

extended to network flow problems and even to gênerai linear programming.A classical example in this case is Kuhn's [15] so called Hungarian method.

Solution procedures for the assignment problem vary from network flows[9, 10, 11], recursive [20], cost parametric [19], relaxation [12, 14], primai dual[6, 15, 17, 21], primai simplex [5, 13, 18] and dual simplex methods [2, 3, 12].

Dual in nature algorithms have (at the present stage of knowledge) betterworst case bounds than primai in nature methods. An efficient version ofKuhn's [15] Hungarian method (see [16, p. 205]), a hybrid algorithm due toBertsekas [6], a relaxation method due to Hung and Rom [14] and threerecently discovered signature methods (Balinski [2, 3] and Goldfarb [12]) haveworst case bound O (n3). All these algorithms are dual in nature. On theother hand, the primai simplex algorithm due to Roohy-Laleh [18] and thealgorithm due to Hung [13], which is based upon Cunningham's [7] strongfeasible trees (see also Barr et al [5], who call such trees alternating pathbases) run in O (n5) and O (n5 In A) time respectively, where A is the différencebetween the initial and the final values of objective. An exception is anefficient version of Balinski's and Gomory's [4] primai algorithm due toCunningham and Marsh [8], which runs in O (n3) time.

The signature idea has been introduced by Balinski [1]. The same authorused this idea later to construct two particularly simple and efficient algo-rithms (Balinski [2, 3]). Both algorithms are dual simplex methods and runin O (n3) time. Balinski [3] reports that preliminary computational studies arevery encouraging for his algorithms. In addition, the method in [3] includesa theory of strong dual feasible trees with remarkable properties.Goldfarb [12] used the signature idea inductively and developed a relaxationO (n3) algorithm, which produces an optimal solution to the k x k problemfrom an optimal solution to the (fc — 1) x(fc —1) problem.

A signature of a basic tree (solution) to the assignment problem is thevector of its column or row degrees. Signature methods work as follows.They start with a dual feasible tree. Then, choosing properly the edge to bedeleted, construct a séquence of dual feasible trees and stop when a tree witha desired signature is found. The desired signature is of type (2, 2, . . ., 2, 1),and quarantees primai feasibility of the tree (Balinski [2, 3]). Since dualfeasibility is kept throughout the computation, the final tree is also dualfeasible and therefore it is an optimal solution to the assignment problem. Itis obvious that dual feasibility is not required throughout the computation.What is really needed, is dual feasibility of the first tree having the desiredsignature.

Recherche opérationnelle/Opérations Research

GENERALIZATION OF DUAL METHOD FOR AP AND TP 2 7 1

Motivated by this observation we investigated whether dual feasibility canbe relaxed. Our effort resulted in a new dual relaxation procedure, whichcan be seen as a generalization of the dual simplex method for network flowproblems. The algorithms constructed in this way, consist of consécutivestages. Each one of these stages is a set of pivots that resemble to the pivotsof the dual simplex method. The algorithms start with a dual feasible treeand assure that the first tree in every stage is also dual feasible. A stage isinitialized by defining properly a set of edges, which are candidates to bedeleted in subséquent itérations. We call this set "candidate set" and theedges in it "candidate edges". The itérations are similar, in a sense, to theitérations of the dual simplex method, particularly in choosing the non-treeedge to enter the basis. When the entering edge is adjoined to the currenttree, it détermines a unique candidate edge, which is deleted to construct theadjacent tree. In that sensé our algorithms look like a primai simplex method;first a non-tree edge is adjoined and then an in-tree edge is deleted. The stageis completed when ail candidate edges are deleted. Intermediate trees in astage are not dual feasible, except possibly in the case of dual degeneracy.When the outcome of the algorithms is a primai feasible tree, then, this treeis an optimal solution. This is because the primai feasible tree cornes outafter the completion of a stage. This is always the case for full dense problems.Otherwise, the algorithms stop when no edge eligible to enter the basis exists,indicating that the problem has no optimal solution. This case may happenwhen sparse problems are considered. When ail stages consist of a singleitération the algorithms coincide with the dual simplex method. An interestingnew monotonie property possessed by the algorithms is that the per unitimprovement of the objective function during the itérations of a stage, isnondecreasing.

In paragraph 3 we combine the dual relaxation procedure with the signatureidea. In particular, the set of candidate edges consists of edges that can bedeleted by Balinski's [3] compétitive dual simplex algorithm. For this reasonthe reader is advised to be aware of the results in [3]. In paragraph 4 weshow that the nxn assignment problem is solved in at most (n— l)(n —2)/2itérations and in at most O (n4) time. The generalization to transportationproblems is presented in paragraph 5. This is accomplished by introducingtwo types of stages. Section 6 describes how sparsity is handled. The algorithmfor transportation problems is easily extended to minimum cost transhipmentproblems. However, we have not been able to use the dual relaxation proce-dure in phase one. That is the reason why the algorithm for transhipmentproblems is briefly described in the last section, where some properties

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272 K. PAPARRIZOS

and ideas for future research are also presented. The next section includespreliminary results.

The main purpose of this paper is to present a new approach to attacknetwork flow problems. We believe that this approach deserves further atten-tion for three reasons.

(1) Since it is in the earliest stage, future research can be initiated. Thealgorithms are very flexible in choosing the candidate set and the edge to bedeleted. From our perspective, the three most interesting open researchproblems are: (i) Can the procedure be generalized to upper bounding net-work flow problems? (ii) Can the procedure be incorporated into a phaseone? (iii) Is there any variant consisting of a single stage? (2) The approachseems promising from the computational point of view. The new monotonieproperty, which accompanies the procedure (at least) as long as it is appliedto network flow problems, seems that it may reduce the average number ofitérations. In addition, ad vaneed techniques similar to those used in [2, 3, 12]that transform signature methods into O (n3) algorithms, can be used toreduce the number of comparisons per itération. Note that the main work ofalgorithms for network flow problems is done on comparisons. Moreover,since the largest the candidate set the faster the algorithms, effectiveness isexpected in large scale problems. (3) The algorithms construct paths to theoptimal solution passing outside the feasible région. In our opinion this factis very important from the theoretical point of view (and probably from thecomputational). It is well known that there are always shortcuts to theoptimal solution passing through the infeasible région. We consider the resultspresented here as the first step towards constructing such algorithms.

2. PRELIMINARIES

Consider the full dense assignment problem

(AP): minSfZjCyXy, s.t. Itjxij=l, E,-xu=l, x„ è 0, iel, jeJ,

where /={1 , 2, . . ., n} is the set of row nodes (in the figures they are drownas circles), J—{1, 2, . . ., n} is the set of column nodes (in the figures theyare drawn as squares) and ctj are real numbers.

A basic solution to (AP) can be equivalently represented by a spanningtree in the bipartite graph G = (I, J, E), where I is the set of row nodes, J theset of column nodes and E = {(i9j): iel, jeJ} is the edge set. A spanningtree of G contains exactly 2 n — 1 edges. Sometimes we will call spanning trees



simply trees or bases. To every spanning tree, T, there are associated theprimai variables xip iel, je J, the dual row variables u = (ul, u2, • . ., un), thedual column variables v = (vu v2, . . -, vn) and the reduced costs, wip iel, je Jdenoted by xu(T), «£(T), Vj(T) and wo.(T) respectively.

Consider first the primai variables. If an edge (i, j)e T, it is basic, otherwisenonbasic. Similarly a primai variable xtj is basic (nonbasic) if (i, j) is basic(nonbasic). The values xu(T) are determined as follows. Set first xij(T) = 0for every nonbasic variable. Then, compute the values of basic variables sothat équations of (AP) are satisfied. It is well known that the values x^ÇT)are integers and easily computed recursively. The values of dual variables areagain easily determined by solving recursively the system of équations

It is easily verified that u^T) and Vj(T) are integer valued whenever all costsctj are integers. With the dual variables at hand the reduced costs are directlycomputed via the relations

wij(T) = cij-ui(T)-vj(T), iel, jeJ

and they are integer valued whenever the costs are integers.

A spanning tree, T, is said to be "primai feasible" if xtj(T) ^ 0 for iel,je J and "dual feasible" if w^T) ^ 0 for iel, je J, Primai simplex algorithmsdecrease (not always strictly) the objective function moving from one primaifeasible tree to an adjacent one. Dual simplex algorithms increase the objectivefunction moving between adjacent dual feasible bases. Primai and dualsimplex algorithms differ among each other in the way the adjacent trees areconstructed. A primai simplex algorithm adjoins first an edge (g, h)£T tothe current primai feasible tree and then deletes an edge (fc, l)eT\J(g9 h)chosen so that the adjacent tree is also primai feasible. On the contrary, dualmethods delete first an edge (fc, J)eT and then adjoin an edge (g, h) chosenso that { r - ( / c , 0 } U (g, h) is a dual feasible tree.

A pivot opération on the current tree, T, consists in constructing theadjacent tree, T, and in Computing the values x^ÇF), ut(T\ VjÇF) andWij(T'). These values are easier computed from the corresponding valuesassociated with T. The procedure is the same for primai and dual algorithms.

Consider first the primai variables. T [J (g, h) contains a unique cycle,O (g, h)9 that necessarily includes edge (g, h) $ T. Since graph G is bipartite,O (g, h) contains an even number of edges. Therefore, we can label adjacentedges in <P(#, h) so that one is "minus" the other "plus". The deleted

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274 K. PAPARRIZOS

edge (k, l) ^ (g, h) belongs in <D(g, h), and it is labeled minus. Clearly,T' — {TU (g, h)} ~ (k, /) is a spanning tree. The values of the primai variablesassociated with the new tree T are given by

xy (T) = x y (T) - £, if (i, j) e O (g9 h) is minus,

= xy (T) + e, if (i, j) e<ï>(g,h) is plus,

= xl7 (T), otherwise,

where e = xH(T). Figure 1 illustrâtes these opérations.

Figure 1. — A spanning tree T; plus-minus labeling of edges in $>(g, h)and values of new primai variables x^iT').

Consider now the remaining variables. T ~ (k, l) consists of two subtrees.The one, T7*, contains the row mode k, the other, Tc, contains column mode /(see Fig. 2). The entering edge is chosen so that geTc and heTr. Setting$ = wgh(T) the values of new dual variables are given by the relations

M,(r)=w«(n i e r ;

These opérations are illustrated in Figure 2. Using relations (1) we caneasily verify that

(2)

y(T) = wv(T)-8, ieT, je T,

= Wy(T) + 8, l e T , 7 6 7*,

= wy (T), otherwise.

A basic solution to (AP), which is primai and dual feasible is an optimalsolution (by the duality theorem of linear programming). The per unit increaseor decrease of the objective function is wgh(T). Also, it is easy to see that thevalue of the objective function changes at each itération by the amountxgh(T')wgh(T). In dual simplex algorithms it is always wgh(T)^0. Conse-quently, an increase of the objective function occurs if the adjoined edge


T f :

GENERALIZATION OF DUAL METHOD FOR AP AND TP

+ 6 +6

D- 6

Figure 2. — A spanning tree T and subtrees Tr, Tc;values of new dual variables; (g, h) adjoined edge, (k, l) deleted

275

: T C

(#, h) is chosen so that xgh(T') ^ 0. Since for the deleted edge (k, l\ which isfirst determined and has label minus, it is xkl(T) = e < 0, (g, h) must bechosen so that it has label minus. This fact is accomplished by choosing(g, h) between the edges (i,j) such that ieTc,jeT\ Now, using relations (2)it is easüy verified that dual feasibility of T is quaranteed by choosing (g, h)via the relation

wgh (T) = min {wy (T) : i e T\ jeT). (3)

To summarize, a dual simplex algorithm deletes first an edge (/c, /), xkl(T) < 0and then adjoins an edge determined by (3). The procedure is repeated andstops when a primai feasible tree is found.

Reasoning similarly we can see that a primai simplex method adjoins anedge (g, h), wgh(T) < Ö, to the current primai feasible tree T and then deletesanother edge (/c> t)eQ)(g, h\ xkl(T) ^ 0, chosen so that (k, ï) has label minusand (g, h) has label plus.

DÉFINITION 1: We say that a pivot opération on T is dual if the deletedand adjoined edges are both minus.

3. THE NON-DUAL SIGNATURE METHOD

Our signature method is a variant of Balinski's [3] compétitive (dual)simplex algorithm. It constructs rooted trees that have precisely the sameprimai structure as Balinski's strong dual feasible bases. The root of the treesis always row node one and it is kept fixed throughout the computation. Theroot directs the edges away from it. If an edge (i, j)e T is directed from rownode i to column node j it is called "odd", otherwise "even". An edge or

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276 K. PAPARRIZOS

node of T is "higher" than a second edge or node (and the second is then"lower") if the first is in the path joining the second to the root.

Given a tree, T, the column signature is the vector a = (al, a2, . . ., an) ofits column node degrees (Balinski [2, 3]). The row signature b = (bl9 b2,bn) of T is similarly defined.

DÉFINITION 2: If T is a rooted tree satisf ying, (i, j) e T and even, impliesxtj(T) ^ 1, and (i,j)eT and odd, and not an edge (1,/) with j of degree 1,implies xtj ^ 0, then Tis said to be a "strong basis".

Recall now that Balinski's [3] strong dual feasible trees are our strong trees(as described here) with the additional condition that they are dual feasible.Consequently the following Lemma 1 holds for strong bases (for the proofsee [3]).

LEMMA 1: If T is a strong basis that contains a column node j of degree atleast 3, then x y (T) ^ — 1 for the unique odd edge (i, j)eT.

The initial tree, T, has row signature b = (n, 1, 1, . . ., 1) and it is con-structed as in other signature methods. It consists of n edges (l,j)eT,vvij(T) = 0 for7e J, found by setting ^ ( T ^ O , vj(T)=^clj; one edge (i, r)eTfor each i # 1, wir(T) = 0, found by setting

u{ (T) = cir -vr(T) = min ; (cy - Vj (T)).

Each stage of the algorithm starts out with a dual feasible strong tree T.Then, some edges of T are chosen in the following manner. Start tracingdown the paths of T starting from the root. The first time a column node, j ,of degree at least three is met, delete the unique edge in the path incident tocolumn node;. Dénote this edge by (i, j) and call it a "candidate edge". Thedeletion of the candidate edge cuts off from the root a subtree Tj which iscalled "candidate subtree". The candidate subtree Tj is removed from T andanother path in the subtree containing the root is traced down determining anew candidate edge and a new candidate subtree. The procedure is repeatedand stops when the subtree including the root contains no odd edge of degreeat least three. Figure 3 illustrâtes the candidate edges and subtrees of a strongbasis. Notice that a candidate edge (Ï, J) is an odd edge in T and that thepath joining column node j to the root contains no other column node ofdegree at least 3.

The procedure described above is only used to détermine the candidateedges and subtrees. No candidate edge is immediately deleted. The choice ofthe edge to be deleted is made after the entering edge has been determined.Dénote by (iu j \ ) , (i2, j 2 ) . . . (zfc, ;fc) and Th, TJ2, . . ., Th the candidate edges


GENERALIZATION OF DUAL METHOD FOR AP AND TP 277

©Figure 3. — A strong basis; (2,7),

(1,6) candidate edges; T6, T7 candidate subtrees.

and subtrees respectively and set T_ = \J Tjr and T+ = T~ T_. At the first

itération of a stage, the adjoined edge (g, h) is determined via the relation

): ieT_JeT+}. (4)

Edge (g, h) is then adjoined to T and the unique cycle O (g, h) is determined.Q>(g, h) contains necessarily a unique candidate edge, say (ia,jj. Candidateedge 0CTS ja) is deleted in order to produce the adjacent tree T, which maynot be dual feasible. At the next itération, the new subtrees T_ = T_ ~ Ty

and T+ = T+{J T^ are computed. TL and T+ are used in (4) to détermine anew entering edge (g\ h'). <f>(g\ h') contains a new unique candidate edge,which is deleted. The procedure is repeated until all candidate edges aredeleted. Then, the next stage of the algorithm starts. The algorithm stopswhen no candidate edge can be found. Observe that Lemma 1 quarantees

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278 K. PAPARRIZOS

that candidate edges exist as long as the current tree contains column nodesof degree at least three.

Notice that for full dense assignment problems there are always edgeseligible for entering the current basis. This is not true for sparse problems.

4. JUSTIFICATION OF THE ALGORITHM

In this section we show that the algorthm finds the optimal solution in atmost (n— l)(n — 2)/2 itérations. In addition to Lemma 1 we will use thefollowing two lemmas from Balinski [3].

LEMMA 2: If a spanning tree contains a unique column node of degree one,then it is primai feasible.

LEMMA 3; If T is a strong basis and a dual pivot is made by deleting anedge (i, j) with x y ( T ) ^ — 1 and j a column of degree at least 3, then theadjacent tree T is also a strong basis.

Our algorithm performs always dual pivot opérations (in the sense ofDéfinition 1). Since the first tree is a strong basis, by Lemma 3 all treesconstructed by the algorithm are also strong bases. Suppose now that thealgorithm stops. Then, there is no candidate edge, meaning that there is noodd edge (i,j) incident to a column node of degree at least 3. Consequently,all column nodes are of degree 2 or 1. Since there are 2n—l edges thealgorithm stops with a strong basis, T, containing a unique column node ofdegree 1 and by Lemma 2, T is primai feasible. Therefore we have shownthe following.

THEOREM 4: The algorithm stops with a primai feasible strong basis.

To prove that the algorithm terminâtes with an optimal solution to (AP)it remains to be shown that the last strong basis is dual feasible. To this endit suffices to show that the first tree in every stage is dual feasible. Recallthat the algorithm stops always at the beginning of a stage.

Let T\ Tk+1 be two consécutive strong bases in a stage of the algorithm.Dénote by Tk the candidate subtrees associated with the strong tree Tk andset

Tk_ = Uj Tkp T\ = Tk ~ Tk_.


GENERALIZATION OF DUAL METHOD FOR AP AND TP 279

Also, set

: ieT\jeTk_}.

DÉFINITION 3; Given a subtree, T', of a spanning tree, T, we say that T' isdual feasible if wu(T) ^ 0 for every edge (i, j) such that ieT\ je T.

LEMMA 5; Let Tk, Tk+1 be two consécutive strong bases in a stage of thealgorithm. IfTk_ is dual feasible and 5fc ̂ 0, then 5fc + 1 ^ ôfc.

Proof: Let T\, be the candidate subtree eut off from Tk. Since

(5)

= min{al5 a2}y

where

: Îe7*_+1,jeT*},

) : ieTk_+\jeTk}.

For i e Tk_+ 1 = Tk_ ^ T\ and ; e Tk+ it is wy (T

k + 1)= wtj (T^. Since Tfc_+1 c Tfc_we conclude that ax ̂ ôfc. Candidate tree T\ contains the column node towhich the deleted edge is incident. Using relations (2) we getwij(T

k + 1) = wiJ(Tk) + Sk for ieTk_+\ jeT\. Since Tk_ is dual feasible,

wy(7*) ^ 0, i ' e r ^ 1 c Tfc_,7eTÎ c Tfc_. Therefore, it is a2 ̂ ôfe and by (5) itis 5k + 1 ^ 5k completing the proof. A

LEMMA 6: Let Tk, /c = l, 2, . . . be consécutive strong trees in a stage o f the

algorithm. IfT1 is dual feasible, then 5'k + i ^ - 8 f t , fe = l5 2, . . .

Proq/; Let T} be the candidate subtree eut off from T1. Then

: ieT2+JeTi}

l5 b2},

where

By relations (2), wl7(T2) = wij-(r

1) for ieT\, je Tl. Dual feasibility of T1

implies that &! ̂ 0. For IGTJ, jeT2 . we get (again) from relations (2) that

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2 8 0 K. PAPARRIZOS

wij(T2)=wij(T

1)-èv Since 6t ^ 0, wu(T2) ^ -6X, î e r J j e T t . Therefore

Assume that the lemma is true for k, k^2. By Lemma 5 and dualfeasibility of T1 we get inductively

0 ^ Sx ̂ S2 ̂ . . . ^ S k ^ 5 k + 1 .

Let T$ + 1 be the candidate subtree eut off from Tft + 1.

Then

where

b'2 = min {wy (Tk+2) ; i G 7 * + \ j G 7*_+ 2 } .

Reasoning as in the case k—l, we get b2 = ~~Sft+1. Since Tk*2 cz T*L+1 andwu(T* + 2) = w0.(T

fc + 1) for ieTk+

+\ jel**2, b{^Sfk+1. By the induction

hypothesis, b\ ^ ô^+1 ̂ — Sfc. Since

0 S 8fc S 5fc+1, S i + 2 = min{6'1, b'2} ̂ - 5 f c + 1 . A

THEOREM 7; T/ie first tree in every stage of the algonthm is dual feasible.

Proof: It suffices to show the statement: if the initial tree in a stage is dualfeasible, the initial tree in the next stage is also dual feasible. The theorem isthen true because the initial tree in the first stage is dual feasible.

Let T1, T2, . . . be the consequitive trees in a stage. We will showinductively that T\ are dual feasible subtrees. Clearly, T\ is dual feasible(because T1 is dual feasible). Assume Tk

+ is dual feasible. ThenTk

++1 = Tk

+ U Ti» where T\ is the candidate subtree eut off from Tfc. By theinduction hypothesis wij(T

k + 1)=wij(Tk) ^ 0 for ieTk

+, je T + . For the edges(ijl ieT\, jeT\y it is wy(7* + 1) = wu(T

k) = Wy(r1) ^ 0. For edges (ij\ieT\jeTk

+y it is wij(Tk+1) = wij(T

k)-§k £ 0. For edges (ij\ ieT\jzT\using Lemma 6 we get

wlV(Tfc+1) = w0.(Tk) + 5fc ̂ ô̂ + 5fc ̂ -8k_1 + 5fc ̂ 0

because wu(Tk) ^ Si for ieT\, jeTk a Tk_ and Sfc ̂ bk_1 (by Lemma 5).

Therefore, for every edge (f, j)9 ieTk+

+\ jeTk+

+1 it is wtj(Tk+1) ^ 0.


GENERALISATION OF DUAL METHOD FOR AP AND TP 2 8 1

Consider now the first tree at the next stage. Since all candidate subtreesof the previous stage have been eut off, it is dual feasible. A

THEOREM 8: The algorithm stops with an optimal solution to the (AP).

Proof: It comes easily from Theorems 4 and 7. A

The counting process for determining the maximum number of itérationsuses the notion of the "level" of a tree. The level of a tree is the number ofI's in its column (or row) signature, Balinski [2, 3]. Using levels Lemma 2can be stated: if a tree is in level 1, it is primai feasible. We will also usethe following notation. T1 is the first tree in level a, 1 ̂ a ^ n—1, andT2, T3, . . ., Tq are the first trees in the next, second next and so on stagesrespectively. Every tree, T\ i—l, 2, . . ., q is in level a and Tq + 1 in levela —1. Notice that if a column node has degree at least two it will neverbecome of degree one. Therefore the level is non-increasing. Let now n(T)dénote the number of column nodes of tree (or subtree) T. Finally, let tt bethe number of the remaining itérations at the first stage and ti9 i = 2, 3, . . ., qthe number of itérations at stage L

THEOREM 9: The algorithm solves the nxn assignment problem in at most

(n -1) (n - 2)/2 itérations.

Proof: If a candidate subtree is eut off and the level is not reduced, theeut off subtree will be part of a candidate subtree at the next stage. This isso because the candidate edges are incident to column nodes of degree atleast three and the degree of column node h is at least two. Therefore,

n(Tl) <n(Tl) < . . . < n(VL).

Clearly, tx ^ n(T\_). For i = 2, 3, . . ., q it is

because the candidate edges at Tl are incident to column nodes not in TL *.

Therefore, the number of itérations, t, at level CT is

q

(TL contains column nodes of degree at least 2). Since 1 ̂ a ^ n— 1 and thealgorithm stops when the first tree in level 1 is reached, the maximum number

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282 K. PAPARRIZOS

of itérations is

1+2+.. . +n-2 = (n-l)(n-2)/2. A

THEOREM 10: (n-l) (n — 2)/2 ÏS t/ie best

Proof: If at each stage there is a unique candidate edge, the algorithmcoincides with Balinski's [3] compétitive (dual) simplex algorithm. Therefore,the algorithm applied to Balinski's [2] assignment problem, defined byCij = (m — i) (n—ƒ), takes precisely (n — 1) (n — 2)/2 itérations. A

T H E O R E M 11: TTie algorithm solves the nxn assignment problem in at most

O (n4) time.

Proof: Clearly, each itération of the algorithm requires at most O (n2) time.The work required in determining the candidate set is done in no more than2n — 1 comparisons. Therefore, the nxn assignment problem is solved in atmost O (n4) time. A

Note 1; The counting tricks used in [2, 3, 12] that transform signaturemethods into O (n3) algorithms don't seem to be combinable with our algo-rithm. The reason is that several stages may be required to reduce the levelby one.

5. AN ALGORITHM FOR FULL DENSE TRANSPORTATION PROBLEMS

Everything said in section 2 can be easily extended to transportationproblems

s, t. Xj xtJ = au Sf xu = bp x y ^ 0, J

where iel={l, 2, . . . , m}, j e J = { l , 2, . . . , n}, S f a ^ £,. bp at^ l, iel, and

bj^lJeJ.A basic solution to (TP) is a tree containing m + n— 1 edges. The values

Xy(T)5 Mj(T),- Uj(T) and wy(T), associated with the tree T, are determined ina similar manner, Le., the values xl7(T) are determined via the relationsxi} (T) = 0 for (i, 7) £ T, the values a, (T), ^ (T) via the relations^ ( T ) + ^(70 = ̂ ( ï ,»eTand Wy(T) = C y - W l . ( ^ ) - ^ ( n

The initial rooted tree (the root is always row node 1) is constructed as inthe algorithm for the (AP). Notice that at the initial tree only odd edges canbe primai infeasible. Ail primai infeasible edges (1, j) are the candidate edges



to start the first stage. The adjoined and deleted edges are chosen as in astage of the algorithm for the (AP). When the first stage is completed eithera primai feasible tree has been constructed and the algorithm stops with anoptimal solution (because the current tree is also dual feasible) or not. If not,a new stage starts.

The algorithm for (TP), unlike the algorithm for (AP), consist of two typesof stages. We call them "odd" or "even". A stage is said to be "odd" (even)if all candidate edges are "odd" (even). The first stage of the algorithm isalways odd. Similarly, a candidate subtree is "odd" (even) if the candidateedge associated with it, is odd (even). The candidate edges are determined ina manner similar to that in the algorithm for the (AP). For instance, thecandidate edges of an odd (even) stage are determined as follows. Starttracing down a path starting from the root; the highest odd (even) edge inthe path, which is primai infeasible, is a candidate edge. Then, choose anotherpath and stop when all paths have been examined.

The pivot opération of an even stage is slightly different than that of anodd stage. Let T be the current tree in an even stage and Tx a candidatesubtree associated with the candidate edge (k, /), kei, leJ. Since now ke 7\the entering edge (g, h) must be chosen so that heTv Suppose (k, l) is deletedand (g, h) is adjoined to produce the next tree T. Setting wffh(T) = 5 andapplying relations (1) it is easily varified that

ieTl9 jeT~Tit

5, ieT~Tu jeTu (6)

otherwise.

The edge to be a adjoined is chosen by the relation

wgh(T) = mm{wij(T): isT+JeT_}. (7)

Using relations (6) and (7) and slightly modifying the arguments of section 4(interchange the rôles of row and column nodes), it can be easily verifiedthat an even stage is completed with a dual feasible tree.

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284 K. PAPARRIZOS

Now, it is clear how the algorithm for the (TP) works. When a stage iscomplétée!, examine whether the current tree is primai feasible. If it is, stop.Otherwise, either an odd or an even stage can be applied to continue.

6. HANDLING SPARSITY

The two algorithms described earlier work provided they are applied tofull dense problems. When dealing with sparsity there are two difficultésthat we must overcome. First, an initial dual feasible tree may not beconstructed as in the full dense case and second, there might be the casewhere no edge eligible to enter the basis is available. We describe how thealgorithm for (TP) is modified to handle sparse transportation problems.

Suppose some of the costs ctj are undefined because the associated edges(i, j) do not exist. Then, approach the problem as follows. Assign very highcosts to not existing edges (ï, j) and make them temporarily admissible. Suchan edge is called "artificial". Then construct the initial dual feasible tree byapplying the algorithm to this transf ormed as bef ore but consider in computa-tion only existing edges. The initial dual feasible tree can be constructedprovided that each row of the cost matrix has at least one (non artificial)edge. Otherwise, there is no primai feasible solution to (TP). These conditionsare easily checked.

The algorithm terminâtes either when a primai feasible solution is foundor when no edge eligible for entering the basis exists. Dénote by T* the lasttree.

LEMMA 12: If T* is a primai feasible tree of the transformed problemcontaining at least one strictly positive artificial edge, then (TP) is (primai)infeasible. Otherwise, T* provides an optimal solution to (TP).

Proof: Clearly, if T* contains positive artificial edges, (TP) is primaiinfeasible.

Suppose T* does not contain any positive artificial edge. Then, either T*does not contain any artificial edge or ail artificial edges of T* are zerovalued. In the former case T* is a primai feasible solution to (TP). Sinceprimai feasibility is achieved at the beginning of a stage, T* is also dualfeasible. Therefore, T* is an optimal solution to (TP). In the latter case,delete all zero valued artificial edges from T*. Then T* décomposes intosubtrees. It is easy to see that every subtree is an optimal solution of a



subproblem of (TP) having as constraints all row and column equalities forwhich the associated nodes are in the subtree under considération. A

LEMMA 13: If T* is not primai feasible and no edge eligible for entering thebasis exists, then (TP) has no optimal solution.

Proof: Case (1). The stage is odd. Let ut, vp wtj be the values of dualvariables and reduced costs associated with T*. We will show that the dualproblem of the transformed problem is an unbounded linear program. Noticethat T* may not be dual feasible. For i e I and j e J we set

U; = tt£ + 8, ie T*L9

= ut, ie T%,

and

= vj, jeT*.

Since there are no edges (i9 j)9 ieT*, jeT% the variables Ui9 Vj constitute adual feasible solution of the transformed problem for sufficiently large 8. Tosee this observe that

wu = ctj - üt - Vj = xv y + 5, for ieT\9je TTt

= wijy otherwise.

It suffices to take 5 ̂ -min{wy; ieT%JeT±}. Then wy + 5 ̂ 0 for ieT%,jeTt and because of dual feasibility of TÎ, T%, it is wtj ̂ 0 whenever i,

The value of the objective function, ƒ (u, v), associated with the dualvariables w, v is given by

Let A be the sum of the candidate variables. Then A < 0. Subtracting allcolumn équations associated with column nodes in T* from row équationsassociated with row nodes in T* results in

Z *t- E, bj=-A>0.i e T - j e T*-

Now, it is easily verified that

f(ü,v)=f(u9v)-5A.

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2 8 6 K. PAPARRIZOS

So, the maximum of the dual objective is unbounded. The dual of (TP) andthe dual of the transformed problem have the same objective function (rowand column équations of (TP) are constraints in the transformed problem).The feasible région of the dual of the transformed problem is contained inthat of the dual of (TP) because the constraints associated with artificialedges do not appear in the constraint set of the dual of (TP). Therefore thedual of (TP) is also feasible and unbounded. This complètes the proof forcase (1).

Case (2). The stage is even. The proof of the Lemma for case (2) is madeby interchanging the roles of T* and T* and following the arguments incase (1). A

The conclusion that the method processes sparse (TP) comes easily fromLemmas 12 and 13. The specialization to (AP) is obvious.

7. CONCLUDING REMARKS

1. The dual relaxation procedure can be easily generalized to minimum costtranshipment problems so long as an initial dual feasible basis is available. Atranshipment problem is a transportation problem with intermediate nodes,i. e., some of the a/s and b/s are zero valued. In addition all edges aredirected. In terms of graphs, the transhipment problem is represented by agênerai (not bipartite) directed graph. The reader acquainted with the dualsimplex method for transhipment problems will have no difficulty in verifyingthat our dual relaxation procedure works provided the candidate edges in astage are chosen so that all are directed either towards the root or awayfrom the root. The construction of the initial dual feasible tree is not thatsimple as it is in transportation and assignment problems. Restoring to aphase one, destroys the graphical représentation of the problem. On the otherhand, adopting a big M method similar to that used for handling sparsityseems more promising.

2. Most of the time in simplex type algorithms for assignment and trans-portation problems is consumed on comparisons that détermine the enteringedge. When our algorithm is used, the number of comparisons required peritération can be substantially reduced if the entering edges in a stage arechosen as follows. For every row node ieT_ set 5(- = min {w^iT): jeT+}and choose the entering edge by the relation wgh(T)=min{5i: ieT_}. Atnext itération for ieT'_ find ô̂ = min{5£, min{wy(7v): 7'eTj} where Tx isthe candidate subtree eut off from T. Then w^(T') = min {5;.* ie TL}. When



this trick is employed the algorithm for (TP) becomes O(kmri) [the signaturemethod becomes O (kn2)], where k is the number of stages. It is thereforedésirable to have k as small as possible. This f act somehow implies that thestages must be large, i. e., the number of candidate edges in each stage mustbe relatively big.

3. There is another reason to believe that the algorithms may perform wellprovided the stages are large. Examine for example, what is happening duringa stage, say an odd stage. At the first itération, it is5 = min{wl7; ieT_JeT+}. If the first candidate subtree T1 is eut off by aclassical dual simplex algorithm the per unit improvement of the objectivefunction is 6/ = mm{wij: ieT^jeT* T^. Since 7\ a T_, 5 ^ S' meaningthat the real improvement of the objective function is larger when ourdual relaxation procedure is employed. Examine also, the effect of the newmonotonie property in subséquent itérations. In particular, the per unitimprovement at the last itération of a stage is larger (probably much larger)than it would be if the same candidate subtree is eut off (from the initial treein the stage) by a purely dual method. Clearly, safe conclusions cannot bederived from such comparisons since different algorithms follow differentpaths to the optimal solution. However, our arguments indicate that the peritération improvement of the objective function is probably (in average) largerwith our algorithms.

4. We described the algorithms in such a way that the deleted edge isalways a candidate edge. However, this is not necessary. The algorithms workprovided (a) the objective function does not decrease and (b) the first tree atthe next stage is dual feasible. It is easy to see that (b) is quaranteed bydeleting any edge belonging to $ (g, h) O T_ and (a) by deleting either aprimai feasible edge with label minus or a primai infeasible edge with labelplus. From the computational point of view it seems promising to enlarge asmuch as possible the set of edges eligible to be deleted. Then, the edge to bedeleted can be chosen to be the best one among all eligible edges. Manycriteria for choosing the best edge can be used. For example choose the edgeinferring the maximal improvement of the objective function. We have alreadybeen able to take care of this problem. Actually, we have constructed analgorithm for the transportation problems consisting of a single stage. Thiswork will be the subject of a fortheoming paper.

5. It is well known that simplex type algorithms for network flow problemsuffer from degeneracy (stalling), particularly when the data are integers.However, there is no known pivoting rule that prevents cycling in the networkdual simplex method analogous to the elegant cycling free rules of the network

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2 8 8 K. PAPARRIZOS

primai simplex methods as it is for example Cunningham's [7] method ofstrong feasible trees. Moreover, it is not quite so obvious that cycling freepivoting rules for gênerai linear programming problems can be easily adaptedto our dual relaxation method. This area seems to be promising for futureresearch.

6. Enlarging the stages seems to be promising from the computationalpoint of view. A question coming to someone's mind is if enlargements canbe done by combining the two types of stages. In other words, does themethod work when the candidate edges in a stage are both even and oddedges? Unfortunately, the answer is no as long as the entering edge is chosenin the way we have described. The reason is that the first tree at the nextstage may be not dual feasible. However, other rules for choosing the enteringedge may work.

REFERENCES

1. M. L. BALINSKI, Signatures des points extrêmes du poiyhèdres dual du problème detransport, C.R. Acad. Sci. Paris, 296, Série I, pp. 456-459, 1983.

2. M. L. BALINSKI, Signature Methods for the Assignment Problem, OpérationsResearch, 33; 1985, pp. 527-537.

3. M. L. BALINSKI, A Compétitive (dual) Simplex Method for the Assignment Problem,Mathematical Programming, 34, 1986, pp. 125-141.

4. M. L. BALINSKI and R. E. GOMORY, A Primai Method for the Assignment andTransportation Problems, Management Science, 10, 1964, pp. 578-593.

5. R. BARR, F. GLOVER and D. KLINGMAN, The Alternating path Basis Algorithm forAssignment Problems, Mathematical Programming, Vol. 13, 1977, pp. 1-13.

6. D. BERTSEKAS, A new algorithm for the Assignment Problem, Mathematical Pro-gramming, Vol. 21, 1981, pp. 152-171.

7. W. H. CUNNINGHAM, A Network Simplex Method, Mathematical Programming,Vol. 11, 1976, pp. 105-116.

8. W. H. CUNNINGHAM and A. B. MARSH, III, A Primai Algorithm for OptimumMatching, Mathematical Programming Study, 8, 1978, pp. 50-72.

9. E. A. DINIC and M. A. KRONRAD, An Algorithm for the Solution of the AssignmentProblem, Soviet Mathematics Doklady, Vol. 10, 1969, pp. 248-264.

10. J. ENDMONDS and R. M. KARP, Theoretical Improvements in Algorithmic Efficiencyfor Network Flow Problems, Journal of the Association for Computing Machinery,Vol. 19, 1972, pp. 248-264.

11. L. R. FORD and D. R. FULKERSON, Flows in Networks, Princeton University Press,Princeton, 1962.

12. D. GOLDFARB, Efficient dual Simplex Methods for the Assignment Problem, Math-ematical Programming, Vol. 33, 1985, pp. 127-203.

13. M. S. HUNG, A Polynomial Simplex Method for the Assignment Problem, Opéra-tions Research, Vol. 31, 1983, pp. 595-600.



14. M. S. HUNG and W. O. ROM, Solving the Assignment Problem by Relaxation,Opérations Research, Vol. 28, 1980, pp. 969-982.

15. H. W. KUHN, The Hungarian Methodfor the Assignment Problem, Naval ResearchLogistics Quarterly, Vol. 2, 1955, pp. 83-97.

16. E. LAWLER, Combinatorial Optimization, Network and Matroids, Holt, Rinchartand Winston, New York, 1976.

17. J. MUNKRES, Algorithms for the Assignment and Transportation Problems, S.I.A.M.Journal, Vol. 5, 1957, pp. 83-97.

18. E. ROOHY-LALEH, Improvement to the Theoretical Efficiency of the Network SimplexMethod, Ph. D. Thesis, Carleton University, 1981.

19. V. SRINIVASEN and G. L. THOMSON, Cost Operator Algorithms for the TransportationProblem, Mathematical Programming, Vol. 12, 1977, pp. 372-391.

20. G. L. THOMSON, A Recursive Method for the Assignment Problems, Annals ofDiscrete Mathematics, Vol. 11, 1981, pp. 319-343.

21. N. TOMIZAWA, On some Techniques Usefull for Solution of Transportation NetworkProblems, Network, Vol. 1, 1972, pp. 173-194.

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